Remark 2.1 
Under Assumptions REF  and REF , for any \(u(\cdot )\in L^2_{\mathcal {F}}(0,\infty ;\mathbb {R}^m)\) , the corresponding state process \(X(\cdot )\)  of (REF ) indeed satisfies both the \(L^2\) -stable condition (see e.g. Definition 2.1 in [1]}):
\(X(\cdot )\in L^2_{\mathcal {F}}(0,\infty ;\mathbb {R}),\) 
